2016Feb 7-13

BędlewoPoland

Up to 30Participants

4-6Speakers

Speakers

Experimental Mathematics in Dynamical Systems

Recent years have seen the flowering of “experimental”
mathematics, namely the utilization of
modern computer technology as an active tool in
mathematical research. This development is not
limited to a handful of researchers nor to a
handful of universities, nor is it limited to one
particular field of mathematics. Instead, it involves
hundreds of individuals, at many different institutions,
who have turned to the remarkable new
computational tools now available to assist in their
research, whether it be in number theory, algebra,
analysis, geometry, or even topology. These tools
are being used to work out specific examples,
generate plots, perform various algebraic and
calculus manipulations, test conjectures, and explore
routes to formal proof. Using computer tools
to test conjectures is by itself a major timesaver
for mathematicians, as it permits them to quickly
rule out false notions.

In these three talks we introduce the use of several computational techniques in the study of
different problems in Dynamical Systems. We briefly detail how the mixed use of
techniques like chaos detection techniques, continuation techniques,
computer-assisted proofs, very high-precision numerics
and specially designed methods permits to locate and study the origin of some interesting
phenomena in Dynamics, as the global organization in neuron models, the global homoclinic
structure of Lorenz-like systems, the coexistence of chaos-hyperchaos behavior,
the existence of safe regions in open Hamiltonian systems, and so on.

Logics for discrete gene regulatory networks

The beginning of the course presents the basic modelling approach defined by
René Thomas (Brussels) in the 70's. We firstly explain how the space of
possible gene expression levels can be decomposed into several intervals in
order to obtain a discrete qualitative description of gene networks. Then, we
show how this discrete approach can be formalized and how the minimal set of
Thomas' parameters is used to build an "asynchronous" automaton from the gene
interaction graph. This automaton models the dynamic behaviour of the
regulatory network (chronological evolution of the gene expression levels).

The main difficulty when modelling gene networks is the identification of the
parameters that govern the dynamics. We show how to use temporal logic in
order to extract unknown parameter values from the observed behaviours. We
firstly give an overview on CTL (Computation Tree Logic) and its associated
model checking methods. We show how it can be used in order to find the set of
possible parameter values, i.e., parameters that are consistent with the known
qualitative behaviours. Some complementary reasonings, taking into account the
biological question under interest, can also help reducing the size of the
gene network model.

We also present a new approach based on Hoare logic and an associated weakest
precondition calculus that generates constraints on the parameter values. Once
proper specifications are extracted from biological traces (e.g. based on
transcriptomic data), they play a role similar to programs in the classical
Hoare logic. The method is correct and complete w.r.t. the Thomas' semantics.

Lastly, we sketch how some logical considerations can be used in order to
generate interesting "wet" biological experiments, starting from the formal
descriptions of the interaction graph and the biological hypotheses under
consideration.

A few biological examples of small size will illustrate the notions defined
during the course.

Topics in Topological Data Analysis

The new emergent area of topological data analysis(TDA) investigates
topological features hidden in data. This requires structural analysis
of various complexes built on top of the data including their
relationship to the ground truth and design of efficient algorithms to
extract these structural information. Persistent homology and
algorithms to compute them are central to this development. In this
course, after introducing the general framework for persistence and TDA
we focus mainly on two topics: (i) Sparsification (ii) Multiscale
mapper. Specifically, we cover the following materials:

Validation and control of hybrid systems

The aim of this course is to introduce the audience to the field of validation (or static analysis) of programs, and hybrid systems and
its relationship to dynamical systems and control theories. Programs
can be modeled as discrete dynamical systems, and hybrid systems are combinations of these discrete dynamical systems with
continuous time dynamical systems (modeled for instance by ordinary differential equations, or differential inclusions).

In the lectures we would cover

The basics of static analysis of programs and hybrid systems : set-based dynamical systems, (polyhedric and semi-algebraic) abstractions,
and calculations of fixed points of such set-based dynamical systems

Proving stability of such systems, in some region of the state space : classical Lyapunov approaches and elaborations of such, and a topological approach based on Wazewski’s property, and algorithms for determining isolating blocks

Viability and controllability of such systems, still using a topological approach on differential inclusions ; some algorithmics.

In this lecture series, we present two rigorous computational methods for enclosing level
curves and level sets of smooth functions, as well as a number of associated applications in
dynamical systems and partial differential equations. The first of these methods consists of
an adaptive randomized subdivision algorithm which can be used to rigorously determine the
topology of sub- and super-level sets of smooth functions, and it will be shown that it can
be extended to find isolating blocks in dynamical systems. In this way, methods from Conley
index can be used to derive global dynamical objects. The second method is centered around a
numerical version of the implicit function theorem, which can be used to verify equilibrium
bifurcation diagrams for a wide variety of evolution equations. This approach will be
illustrated in the context of concrete examples from high-dimensional lattice dynamical
systems, as well as for the infinite-dimensional diblock copolymer model in materials
sciences.

Schedule

Sunday (7th February 2016) is an arrival day. Lectures will start on Monday morning.
Winter school will end with a lunch around noon on Saturday (13th February, 2016).

Afternoons will be left for private discussions, collaboration and
small mini-workshops. We do not plan any contibuted talks.

Friday

Saturday

Meals

Meals will be served at

8.00-9.00

Breakfast

11.00-11:30

Coffe break

13.00

Lunch

19.00

Dinner

Registration & Pricing

Participation in Winter School on Computational Mathematics
is subject to personal invitation.

Young researchers interested in participation,
in particular PhD students and post-docs, can contact organizers by email.
Please provide your research interests and short recommendation letter
from your scientific advisor.
We anticipate that we will be able to offer some number of grants
to cover registration fee.
If you are interested in financial support please indicate this when sending the above email.

The registration fee is 800 PLN. It covers accommodation and full board.
According to the current exchange rates (as of November 26, 2015) 800 PLN
is equivalent to 190 EUR or 200 USD.
For the details on how to pay see the registration fee Event FAQs below.

Event FAQs

From Poznań the best option is to take a taxi.
To avoid excessive taxi fares (even up to 400PLN) we recommend booking a taxi in advance
through conference center in Będlewo. Please contact them by email (bedlewo@impan.pl)
providing flight details and mobile number. The cost is 110 PLN and the taxi can be shared
by up to 3 people. Taxi driver will wait at the airport (close to cash exchange office) or
at the railway station (close to ticket office no. 1).